Trigonometry Examples

${(3 (\cos (15) + i \sin (15)))}^{3}$

Step 1

Simplify each term.

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Step 1.1

The exact value of $\cos (15)$ is $\frac{\sqrt{6} + \sqrt{2}}{4}$ .

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Step 1.1.1

Split $15$ into two angles where the values of the six trigonometric functions are known.

${(3 (\cos (45 - 30) + i \sin (15)))}^{3}$

Step 1.1.2

Separate negation.

${(3 (\cos (45 - (30)) + i \sin (15)))}^{3}$

Step 1.1.3

Apply the difference of angles identity $\cos (x - y) = \cos (x) \cos (y) + \sin (x) \sin (y)$ .

${(3 (\cos (45) \cos (30) + \sin (45) \sin (30) + i \sin (15)))}^{3}$

Step 1.1.4

The exact value of $\cos (45)$ is $\frac{\sqrt{2}}{2}$ .

${(3 (\frac{\sqrt{2}}{2} \cos (30) + \sin (45) \sin (30) + i \sin (15)))}^{3}$

Step 1.1.5

The exact value of $\cos (30)$ is $\frac{\sqrt{3}}{2}$ .

${(3 (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \sin (45) \sin (30) + i \sin (15)))}^{3}$

Step 1.1.6

The exact value of $\sin (45)$ is $\frac{\sqrt{2}}{2}$ .

${(3 (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \sin (30) + i \sin (15)))}^{3}$

Step 1.1.7

The exact value of $\sin (30)$ is $\frac{1}{2}$ .

${(3 (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2} + i \sin (15)))}^{3}$

Step 1.1.8

Simplify $\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}$ .

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Step 1.1.8.1

Simplify each term.

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Step 1.1.8.1.1

Multiply $\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2}$ .

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Step 1.1.8.1.1.1

Multiply $\frac{\sqrt{2}}{2}$ by $\frac{\sqrt{3}}{2}$ .

${(3 (\frac{\sqrt{2} \sqrt{3}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2} + i \sin (15)))}^{3}$

Step 1.1.8.1.1.2

Combine using the product rule for radicals.

${(3 (\frac{\sqrt{2 \cdot 3}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2} + i \sin (15)))}^{3}$

Step 1.1.8.1.1.3

Multiply $2$ by $3$ .

${(3 (\frac{\sqrt{6}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2} + i \sin (15)))}^{3}$

Step 1.1.8.1.1.4

Multiply $2$ by $2$ .

${(3 (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2} + i \sin (15)))}^{3}$

Step 1.1.8.1.2

Multiply $\frac{\sqrt{2}}{2} \cdot \frac{1}{2}$ .

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Step 1.1.8.1.2.1

Multiply $\frac{\sqrt{2}}{2}$ by $\frac{1}{2}$ .

${(3 (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{2 \cdot 2} + i \sin (15)))}^{3}$

Step 1.1.8.1.2.2

Multiply $2$ by $2$ .

${(3 (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} + i \sin (15)))}^{3}$

Step 1.1.8.2

Combine the numerators over the common denominator.

${(3 (\frac{\sqrt{6} + \sqrt{2}}{4} + i \sin (15)))}^{3}$

Step 1.2

The exact value of $\sin (15)$ is $\frac{\sqrt{6} - \sqrt{2}}{4}$ .

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Step 1.2.1

Split $15$ into two angles where the values of the six trigonometric functions are known.

${(3 (\frac{\sqrt{6} + \sqrt{2}}{4} + i \sin (45 - 30)))}^{3}$

Step 1.2.2

Separate negation.

${(3 (\frac{\sqrt{6} + \sqrt{2}}{4} + i \sin (45 - (30))))}^{3}$

Step 1.2.3

Apply the difference of angles identity.

${(3 (\frac{\sqrt{6} + \sqrt{2}}{4} + i (\sin (45) \cos (30) - \cos (45) \sin (30))))}^{3}$

Step 1.2.4

The exact value of $\sin (45)$ is $\frac{\sqrt{2}}{2}$ .

${(3 (\frac{\sqrt{6} + \sqrt{2}}{4} + i (\frac{\sqrt{2}}{2} \cos (30) - \cos (45) \sin (30))))}^{3}$

Step 1.2.5

The exact value of $\cos (30)$ is $\frac{\sqrt{3}}{2}$ .

${(3 (\frac{\sqrt{6} + \sqrt{2}}{4} + i (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \cos (45) \sin (30))))}^{3}$

Step 1.2.6

The exact value of $\cos (45)$ is $\frac{\sqrt{2}}{2}$ .

${(3 (\frac{\sqrt{6} + \sqrt{2}}{4} + i (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \sin (30))))}^{3}$

Step 1.2.7

The exact value of $\sin (30)$ is $\frac{1}{2}$ .

${(3 (\frac{\sqrt{6} + \sqrt{2}}{4} + i (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2})))}^{3}$

Step 1.2.8

Simplify $\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}$ .

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Step 1.2.8.1

Simplify each term.

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Step 1.2.8.1.1

Multiply $\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2}$ .

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Step 1.2.8.1.1.1

Multiply $\frac{\sqrt{2}}{2}$ by $\frac{\sqrt{3}}{2}$ .

${(3 (\frac{\sqrt{6} + \sqrt{2}}{4} + i (\frac{\sqrt{2} \sqrt{3}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2})))}^{3}$

Step 1.2.8.1.1.2

Combine using the product rule for radicals.

${(3 (\frac{\sqrt{6} + \sqrt{2}}{4} + i (\frac{\sqrt{2 \cdot 3}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2})))}^{3}$

Step 1.2.8.1.1.3

Multiply $2$ by $3$ .

${(3 (\frac{\sqrt{6} + \sqrt{2}}{4} + i (\frac{\sqrt{6}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2})))}^{3}$

Step 1.2.8.1.1.4

Multiply $2$ by $2$ .

${(3 (\frac{\sqrt{6} + \sqrt{2}}{4} + i (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2})))}^{3}$

Step 1.2.8.1.2

Multiply $- \frac{\sqrt{2}}{2} \cdot \frac{1}{2}$ .

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Step 1.2.8.1.2.1

Multiply $\frac{1}{2}$ by $\frac{\sqrt{2}}{2}$ .

${(3 (\frac{\sqrt{6} + \sqrt{2}}{4} + i (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2 \cdot 2})))}^{3}$

Step 1.2.8.1.2.2

Multiply $2$ by $2$ .

${(3 (\frac{\sqrt{6} + \sqrt{2}}{4} + i (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4})))}^{3}$

Step 1.2.8.2

Combine the numerators over the common denominator.

${(3 (\frac{\sqrt{6} + \sqrt{2}}{4} + i \frac{\sqrt{6} - \sqrt{2}}{4}))}^{3}$

Step 1.3

Combine $i$ and $\frac{\sqrt{6} - \sqrt{2}}{4}$ .

${(3 (\frac{\sqrt{6} + \sqrt{2}}{4} + \frac{i (\sqrt{6} - \sqrt{2})}{4}))}^{3}$

Step 2

Combine fractions.

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Step 2.1

Combine the numerators over the common denominator.

${(3 \frac{\sqrt{6} + \sqrt{2} + i \sqrt{6} - i \sqrt{2}}{4})}^{3}$

Step 2.2

Combine $3$ and $\frac{\sqrt{6} + \sqrt{2} + i \sqrt{6} - i \sqrt{2}}{4}$ .

${(\frac{3 (\sqrt{6} + \sqrt{2} + i \sqrt{6} - i \sqrt{2})}{4})}^{3}$

Step 3

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 3.1

Apply the product rule to $\frac{3 (\sqrt{6} + \sqrt{2} + i \sqrt{6} - i \sqrt{2})}{4}$ .

$\frac{{(3 (\sqrt{6} + \sqrt{2} + i \sqrt{6} - i \sqrt{2}))}^{3}}{4^{3}}$

Step 3.2

Apply the product rule to $3 (\sqrt{6} + \sqrt{2} + i \sqrt{6} - i \sqrt{2})$ .

$\frac{3^{3} {(\sqrt{6} + \sqrt{2} + i \sqrt{6} - i \sqrt{2})}^{3}}{4^{3}}$

Step 4

Raise $3$ to the power of $3$ .

$\frac{27 {(\sqrt{6} + \sqrt{2} + i \sqrt{6} - i \sqrt{2})}^{3}}{4^{3}}$

Step 5

Raise $4$ to the power of $3$ .

$\frac{27 {(\sqrt{6} + \sqrt{2} + i \sqrt{6} - i \sqrt{2})}^{3}}{64}$